Force of Nature instead of Shurelya's
If you need more MR, sell the Philosopher's Stone for a Force of Nature instead of making it into a Shurelya's.

Skills

Bandage Toss
Amumu's skillshot stun. Important for chasing, support, and can (very rarely) be used as an over-the-wall get-away. Learn to aim this well.

Despair
Amumu's anti-HPstacker skill. This is amazing against everyone, but even more so against high-HP enemies. It is also very useful for Lizard, Golem, Baron, and Dragon.

Tantrum
Amumu's main skill. Passively makes him tougher, and actively great for farming and killing. Learning to hurt multiple minions evenly so you can get many kills at once with Tantrum is very helpful.

Curse of the Sad Mummy
The icing on the cake. Curse is probably the most valuable skill in LoL for team fights when used properly. If you're not initiating, which you should be, you can easily just Flash into the middle of all the enemy champions and pop Curse to snare all of them.

Play-style

Early
Conservative farming is what Amumu is all about at the beginning. Don't be afraid to use your Tantrum just to kill-shot 1 minion. You need all the cash you can manage.

Middle
Still should be farming, but you also need to be helping teammates out with being a gank initiator and/or lane helper.

Late
You should be a godly tank by now with your cape, Veil, and GA/Force of Nature. You also should be destroying people while they (if they're good) focus on your carry(ies) and/or casters: stunning, Tantrum spamming, sunfire degen, Despair degen, Curse if there's 2 or more of them.